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Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups

Ido Grayevsky, Gabriel Pallier

Abstract

We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. We employ our product theorem to distinguish up to quasiisometry certain families of solvable groups which share the same dimension, cone-dimension and Dehn function; actually we do this by distinguishing them up to sublinear bilipschitz equivalence, which is slightly stronger. As an application, we recover the fact, recently obtained by Bourdon and Rémy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.

Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups

Abstract

We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. We employ our product theorem to distinguish up to quasiisometry certain families of solvable groups which share the same dimension, cone-dimension and Dehn function; actually we do this by distinguishing them up to sublinear bilipschitz equivalence, which is slightly stronger. As an application, we recover the fact, recently obtained by Bourdon and Rémy with different groups, that there exists uncountably many quasiisometry classes of indecomposable, non-unimodular, high rank solvable Lie groups.
Paper Structure (18 sections, 26 theorems, 42 equations, 1 figure, 2 tables)

This paper contains 18 sections, 26 theorems, 42 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Acknowledgments
  5. The product Theorem \ref{['thm:KKLgeneralized']}
  6. Preliminaries
  7. Going through cones
  8. Proof of Theorem \ref{['thm:KKLgeneralizedNEW']}
  9. Step $1$.
  10. Step $2$
  11. Step $3$
  12. Proof of Theorem \ref{['thm:KKLgeneralized']}
  13. Completely solvable groups, Corollary \ref{['cor:KKL-symmetric']} and Theorem \ref{['cor:KKLgeneralized']}
  14. Some preliminaries on Cornulier's reduction
  15. Warm-up: the groups in Example \ref{['exm:four-dim-KKL']} are not quasiisometric.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G$ and $G'$ be quasiisometric simply connected nilpotent Lie groups. Then $\rho_\infty(G)$ and $\rho_\infty(G')$ are isomorphic.

Figures (1)

  • Figure 1: Logical dependencies between the main statements in this paper.

Theorems & Definitions (79)

  • Theorem 1.1: PanCBNPansuCCqi
  • Definition 1.2: Commuting up to sublinear error
  • Theorem A
  • Corollary B
  • Definition 1.3
  • Theorem C
  • Theorem 1.4: KKL, Theorem 5.1
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 69 more